Discrete-time modeling of Hamiltonian systems

Abstract

Abstract:The problem of discrete-time modeling of the lumped-parameter Hamiltonian systems is considered for engineering applications. Hence, a novel gradient-based method is presented, exploiting the discrete gradient concept and the forward Euler discretization under the assumption of the continuous Hamiltonian model is known. It is proven that the proposed discrete-time model structure defines a symplectic difference system and has the energy-conserving property under some conditions. In order to provide alternate discrete-time models, 3 different discrete-gradient definitions are given. The proposed models are convenient for the design of sampled-data controllers. All of the models are considered for several well-known Hamiltonian systems and the simulation results are demonstrated comparatively.

Interconnection through u-average passivity in discrete time

“…Remark 3.3: Theorem 3.2 defines a family of feedback interconnections preserving u-average passivity of the overall system. As it is clear from (21), under an external source v and for a fixedū in (16), the average outputs are defined starting from the same outputs of the single systems (8) but averaged over the new interconnected dynamics (20) deduced from (15). Thus, starting from average passivity of the single systems (8) with outputs h i (x i ), the interconnected system is u-average passive under the power preserving stated feedbackū =ū(x) solving (16).…”

“…Let assume the interconnection between two nonlinear van der Pol oscillators in discrete time described by [20], [21],…”

Adaptive interconnection and damping assignment passivity‐based control for linearly parameterized discrete‐time port controlled Hamiltonian systems via I&I approach

“…In order to obtain the discrete‐time modeling of port controlled Hamiltonian systems, we use the method presented in the work of Yalcin et al 49 In this method, the time derivatives are replaced by their forward‐Euler approximations, and the gradient terms are replaced by some discrete gradients that satisfy conditions defined by Gonzalez et al 50 We restate these conditions below.Definition Let H ( x ) be a differential scalar function in x ∈ R n , then is a discrete gradient of H if it is continuous in x and, …”

Discrete port-controlled Hamiltonian dynamics and average passivation

“…The energy-balance (12) emphasizes that the total stored energy can be exactly split into the naturally dissipated and supplied components. This is not the case when considering the discrete pcH form more commonly adopted in the literature following approximate or sampling inspired approaches [6]- [8], [21]. Let us discuss more in detail this aspect which directly affects energy control design.…”

“…x partial supplied energy (21) where by construction the dissipated energy includes internal dissipation (due to control-free dynamics) and an inputdependent component. It results that (21) deduced from (19) does not emphasize the total energy supplied by the external source and this enlightens a critical aspect when dealing with energy-inspired control strategies.…”